Old and New Identities for Bernoulli Polynomials via Fourier Series
The Bernoulli polynomials Bk restricted to [0,1) and extended by periodicity have nth sine and cosine Fourier coefficients of the form Ck/nk. In general, the Fourier coefficients of any polynomial restricted to [0,1) are linear combinations of terms of the form 1/nk. If we can make this linear combi...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2012-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2012/129126 |
| Summary: | The Bernoulli polynomials Bk restricted to [0,1) and extended by
periodicity have nth sine and cosine Fourier coefficients of the form Ck/nk. In general, the Fourier coefficients of any polynomial restricted to [0,1) are
linear combinations of terms of the form 1/nk. If we can make this linear
combination explicit for a specific family of polynomials, then by uniqueness
of Fourier series, we get a relation between the given family and the Bernoulli
polynomials. Using this idea, we give new and simpler proofs of some known identities
involving Bernoulli, Euler, and Legendre polynomials. The method can also be
applied to certain families of Gegenbauer polynomials. As a result, we obtain
new identities for Bernoulli polynomials and Bernoulli numbers. |
|---|---|
| ISSN: | 0161-1712 1687-0425 |